C3 On the Axisymmetric Problem of Elasticity Theory

for a Medium with Transverse Isotropy

^ by

B. A. Bubanks and E. Sternberg

A Technical Report to the

Office of Naval Research Department of the Navy Washington 2$, D. C.

Contract N7onr-32906

Project No. NR 035-302

Department of Mechanics Illinois Institute of Technology

Chicago 16, Illinois

July 1, 1953

On the Axisymmetrlc Problem of Elasticity Theory

for a Ifediisn with Transverse Isotropy

by

R. A. Eubankc and E. Sternberg

Illinois Institute of Technology Chicago 16, Illinois

1* Introduction

The linear theory of elasticity of anisotropic media has long been

the subject of numerous investigations. In particular, the problems of

torsion and flexure of beams for various underlying anisotropic stress-

strain relations, as well as the plane problem for an orthotropic

medium, have been treated extensively in the literature which now contains

a wealth of significant solutions of such problems.

The abandonment of the assumption of elastic isotropy in three-

dimensional problems, on the other hand, leads to greater analytical com-

plications. Here, apparently, only the case in which the material

possesses an axis of elastic symmetry ("transverse isotropy" )» as

illustrated by crystals of the hexagonal system, has received successful

attention* The first investigation of this kind is due to Michell [2J

(1900), who obtained the solution corresponding to a transversely iso-

tropic semi-infinite elastic body under arbitrarily prescribed surface

tractions on its plane boundary.

The results communicated in this paper were obtained in the course of an investigation conducted under Contract N7onr-32906 with the Office of Naval Research, Department of the Navy, Washington 2 5, D. C

A bibliography of two-dimensional investigations based on aniso- tropic stress-strain laws, is beyond the scope of the present paper.

^his terminology is used by Love QQ , p. l60. Numbers in brackets refer to the bibliography at the end of the paper.

Further progress in this connection was made by Lekhnitsky [£) (19b0)»

£ixj» who stated a stress-function approach to problems characterized by

torsionless axisymmetry for the case in which the axis of elastic symmetry

coincides with the axis of stress symmetry. Lekhnitsky's solution of the

displacement equations of equilibrium in terms of a single stress function

satisfying a fourth-order partial differential equation, is a generaliza-

tion of Love's approach^ to isotropic rotationally symmetric problems,

which in turn may be regarded as a specialization of Galerkin's £6[]

general solution of the three-dimensional field equations. Lekhnitsky

£3] employed his approach in securing several technically Important par-

ticular solutions.

Subsequently, Ellict £7] (1°U8) gave a solution of the three-

dimensional field equations for a transversely isotropic medium in terms

of two stress functions, each satisfying a second-order partial differential

equation. This solution, in the special instance of rotational stress

syntetry about the axis of elastic symmetry, reduces to that originated

by Lekhnitsky £3^. Elliot's approach, which is evidently incomplete in

the absence of axisymmetry cf the stress field, has been applied by him

DO* DO and by Shieid £?1 t0 the solution of additional significant

special problems with and without rotational symmetry. Finally, we refer

to a recent paper by Ana Moisil £10^] (1950), in which a formal scheme due

to Gr. C. Moisil p-l^ is used to extend Galerkin's J^T] stress-function

approach to media with transverse iso^ropy. The method of derivation

employed here does not assure the completeness of the approach which

^See [l], p. 27U.

^hi3 observation, according to Mindlin QfJ, is due to H. M. Wester- gaard.

involves three stress functions, each satisfying a partial differential

equation of the sixth order. No attempt is made to effect a reduction

in the unnecessarily high order of the equations characterizing the stress

functions, which is an essential disadvantage from the point of view of

applications, and the case of axieymmetric stress fields is not considered

especially.

It is the purpose of the present paper to present a systematic

derivation of, and to supply the apparently still missing completeness

proof for, Lekhnitsky's approach to problems with torsionless arLsymmetry

in a medium possessing transverse isotrcpy. To this end we first general-

ize a theorem of Almansi Tl2j concerning the representation of the

On general solution of the equation V F = 0 in terms of harmonic

functions. The theorem established here may be of interest beyond the

present application. The material in Sections 3, h, although primarily

expository, has briefly been included in order to render the present

paper sensibly self-contained.

2. A Generalization of Alraansi's Theorem

In this section we prove the following theorem:

Let R be a region of the (p,z)-plane such that a straight line

parallel to the z-axis intersects the boundary of R in at most two

points* Let F (p,z) be a solution of

1 rV?F • V*vl ....^ .V<T =0 in R (2.1) . _ , in 12 n—1 n n

where

v?-A*.^

A = ^/1<f> *

}• (2.2)

¥' z oz

the c. are (not necessarily real) constants, and the functions p. are

continuous in R with pr ^ 0. Then F admits the representation.

(n) Fn(f,z) = Fn_1(y0,z) + zm ¥{p,z) (2.3)

where,

n - 1 (n)

i = l = 0,

and m (0 ^ m £ n-1) is the number of the coefficients c. (i

n- 1) which are equal to c •

For future reference, we first note the trivial identities

V\ [A(z)B(p,z)] = A^B * c^U MB + 2A^z),

9 (i) ? ? (i) . (i)

V =(cJ-Ci)F,zz if ViF = °>

(2.U)

— 1, 2, ••• •

(2.5)"

(2.6)

Subscripts preceded by a comma denote partial differentiation with respect to the argument indicated.

—» o and observe that the product of any two among the operators D ,-A, V.,

2 V. is commutative.

As a further preliminary, we show that if F(e,z) satisfies Vf? = 0

and m > 0 is an integer, then

r

vfcA) =« 0 for k = m - 1

^IncflfF for k = a (2.7)

Let k = m - 1. Clearly, (2.7) holds for m = 1. We proceed by induction.

Thus, assune

2(m-l) (j*-2p):s0#

Then,

VfV-1?) m v2(m-l)72(zm-0T)

= (m-l)ci2vfn-1) [(m^)^^ • 22

m-2Fa[]= 0,

by hypothesis and (2.5). Next, let k = a. Again, (2.7) evidently holds

for m = 1. Using induction once more, assume

^(m-D^m-L^ = gm-l 1^ 2(m-l) ^- 1 ? 1 ' 1 Z

Then,

= mci2Vi

2(m-1)[(m - l)zm-2 F + 2zm-1 FJ

= 2* [» cf If F,

by hypothesis, (2.5), and the result for k = m - 1. This completes the

proof of (2.7).

Turning to the proof of the theorem, we need to show that there (n)

exists a function F(p,z) which satisfies

V2 F(n) = 0, n

n - 1

TTr^fc-1^]"0-

(2.8)

(2.9)

Moreover, we may assume without loss in generality that

2 2 ci = cn

2/2 ci*cn

(i = n-m, n-m + 1, ...,n- 1),

(i = 1, 2, ..., n - m - 1). J

(2.10)

In view of (2.10) we may write (2.9) as

n - 1 n-m-1

rr vfc,=-TT v\ h? [.- ^ and (2.11), by virtue of (2.6), (2.7), and (2.8), becomes,

n- 1 n-m-1

(2.11)

jT V« »„ = * t Jtf — 2 F<»> TT (•} - 4). (2.12) (n)

Thus, it suffices to construct a function F(p,z) which satisfies (2.8)

and (2.12). To this end let

'//%-l „r2n -f m 4 2

g(]3>z) = 1 = 1

n-m- 1

?iFn (2.13)

^cfTTcc^) „-i

the operator 0 being defined by

rf-l„i D~XG(p,z) = 0(JD, £) d£, (2.11)

where z is the value of z belonging to one of the points of inter-

section of a straight line parallel to the t-axLs with the boundary of

R. Evidently, F^n' = g satisfies (2.12). Furthermore,

if—* ?!•-<> (2.15)

because of (2.1). Hence,

2n-m-3 Vng" jEj '"W-

Now, let g#(o,z) be defined by

2n-m-3

Ah2n-m-3 " f2n-m-3,

(2.16)

Ah2n-m-U" f2n-m-li

Ab^ = fk - c* (k + l)(k + 2)hk + 2 (k = 0,l,...,2n-m-5),

The function g^ so determined has the properties

-(2.17)

Vi «. = K&' -f—2«.-o. (2.18)

It follows from (2.13), (2.18) that

F(D) = g - g# (2.19)

satisfies both (2.8) and (2.12). This completes the proof of the theorem.

Successive applications of the foregoing theorem at once yield a

representation of any solution of

i = 1 VI F = 0 (2.20)

in terms of solutions F ' of the equations (i)

8

V [ F(i) = 0 (i = 1, 2, ..., n). (2.21)

In particular, if all cr (i = 1, 2, ..., n) are distinct^

n

X F = > F(k), (2.22)

•whereas if all c, coalesce,

n

F = N 7 zk F(k). (2.23)

In connection with the plane problem of elasticity theory for an

orthotropic medium, special interest is attached to the case in which

(2.2) assume the form,

2 Here p and z are Cartesian coordinates, and \7 i3 a modified two-

dimensional Laplacian operator. In the rotationally symmetric problem

appropriate to a medium with transverse isotropy, on the other hand, we

encounter the case in which

*.32 ia _2_ a2 . i a . 2 a2 f0 90 A = —* + - —- * Vi = 7-5 + • c. —-* , (2.25) ay j9 3yo * ay p ay> i 3z^

2 where p and z are circular cylindrical coordinates and V.. is a

modified axisymmetric Laplacian operator.

Finally, if v\ ifi (2*20) is defined as in (2.2U) or (2.25), and 2

c. = 1 (i = 1, 2, ..., n), then (2.23) becomes identical with Almansi's

Ql2|] representation of the general solution of V *F = 0 in terms of

2 harmonic functions, for the case in which r^ is the two-dimensional or

the axisymmetric three-dimensional Laplacian operator. This special in-

stance arises in the well known applications of Almansi's theorem to the

plane and to the rotationally symmetric problem in the classical theory

of elasticity of isotropic media.

3. Stress-Straiu Relations and Strain-Energy Function for a Medium

with Transverse Isotropy

Let us, temporarily, adopt the notation,

ri = fxK> r2 - ^ *3 = Y*z> fh = ^' r$ = 'zx' ^6 = Ky'>

l xx' 2 yy* 3 zz* u yz* 5 zx* o xy

(3.D

Here, 'f . ... and e , ... are the Cartesian components of stress XX XX

and "infinitesimal" strain, the strains being defined by

exx = \,x> •••• eyz = uy,z + uz,y> — (3'2)

where u , u , u are the Cartesian scalar components of the displace-

ment vector.

The general linear stress-strain law now assumes the form,

*i • cij *y (3*3)

in which i,j range over the integers one to six, and the usual summation

convention is employed. For a homogeneous medium the c. . are constants.

A necessary and sufficient condition for the existence of a strain-

energy function W(e,, ..*, e^), such that

A = gj <3.M is that

cJ±=cir (3.5)

Moreover, \

5 W = ? Ci 1 ei ej * ^3*6^

Note that the present definition of the elastic constants c. . coincides with thai adopted by Love Til, p. 99. ^ ':

9 i!

We now Impose the condition (transverse isotropy) that the stress-strain

law (3.3), (3.5) be invariant under an arbitrary rotation about the

z-axis. This is the case if and only if

ci;J = 0 (i = 1,2,3; 1 = U,5,6), (i = U,5,Cj J t i)

cll = c22» °13 = C23' CUi = Ctt> 2C66 = cll " c12*

•(3.7)

With the notation,

cll = a» c33 = *' Chh */** c66 V*

c^ = a - 2/7, c13 = b,

(3.8)

the stress-strain law (3*3), (3*5), subject to (3.7)> (3.8), becomes

fir = ae~ + (a - 2/T) a^. • be., 'xx xx ir yy zz

fyy=aeyy + (a " 2A>exr + be zz

7i,s b(e + e ) • ae zz xx yy' zz

^yz = /*eyz' ^zx = /<ezx' ^xy V V

(3.9)

whereas (3*6) appears as

10

2W = a(e^ + ejy) + Se^ + 2(a - 2/4,) e e xx yy

+ 2b(e + 0©w„ +/^(QL • elj +M< xx yy zz ~ yz zx /* 3 (3.10) yy zz /^ yz zx' /— xy~

The stress-strain relations (3*9) involve five independent elastic con-

stants, of which JX and ju[ have an obvious physical meaning.

Inverting the first three of (3*9), we reach

11

where

eyy = 1 \?» • * *KX " 7rz •]

e .If/ - 3 (/ • / )1 zz g L zz IT wxx yy J*

_ k/ljaa. - b - a/l) g _ aa - b - a/£

(3.11)

aa - b a -/"

aa - b aa - b

(3.12)

Furthermore,

E= 2^(1 + j)) (3.13)

The physical significance of the four new elastic constants defined in

(3,12) is immediate from (3.11).

A trivial computation, based on (3»10), yields the following neces-

sary and sufficient conditions for the positive definiteness of W:

a > 0, a > 0, u > 0, ju > 0, aa - b2 - a/7 > 0, (3.1U)

or, equivalently,

E > 0, E > ot/U > o,/7 >0, -!<>>< 1, l->)>2 £J-. (3.15)

In the special case of isotropy, we have

E

._ Z_JU7>

M V* = 2(1+ 5T# (3.16)

U. Basic Equations for a Medium with Transverse Isotropy In the Case of

Torsionles3 Axi symmetry

With reference to cylindrical coordinates (p, 0, y), where

x = o cos y, y = p sin y, z = z. (1.1)

the displacement field in the presence of torsionless axisynanetry about

the z-axis, is characterized by

The associated field of strain now becomes

e = u .....i. ZZ ^z'

e = u + u , pz JO,Z z,p» eyj>=ezy=0>

and, according to (3»9)» the stress-strain relations are

/yy=ae/y+(a-2j)ey>+bezz

(U.2)

(U.3)

*i« = 5ezz + b(eyc)o + V

^>z =/">*> ^y/,= ^z=0'

The stress equations of equilibrium, in the absence of body forces, by

virtue of (h»2)f (li.3), and (b.U) here reduce to

(WO

r + r + &—#=o

yoz^o 'zz,z jo

> (U.5)

12

13

and substitution of (U.2), (h.3)> (U.U) into (h.5) yields

ae -f UM + (LL -f b - a)u = 0, j^o / />>zz / z>pz

(/, + b)e>z +/^|^u ) - (^ * b - 5)u2jZZ = o, r r i

tt.6)

is which

e = e J3P

1 9 + v*•« 7 ^ ^>>+ u*.*- (k.7)

5. Derivation and Completeness of Lekhnitsky's Stress-Function Approach

Suppose u (p,z) and u (o,z) satisfy the equilibrium equations (U.6)

in a region R of the meridional half-plane o - 0, -oo < z < co, and let

R have the property that straight lines parallel to the coordinate axes

intersect the boundary of R in at most two points. We may then define

two functions U(p,a) and 7(c,z) by means of

u = U , u = V P $p* Z ,Z

For convenience, we introduce the operators

do ode p dp op

D «±. 2 dz

(5.1)

(5.2)

h (5,3)

V being the axisymmetric Laplacian. Substitution of (5.1) in (li.6),

with the aid of (1.7) and (5.2), yields

— [a V* U + //r^J + {/J. + b) D£ vl = 0

^ [(/£ • b) V*V +JULVl V * iD^] = 0,

or, since (5.1) determine U and V only within arbitrary additive

functions of z and p, respectively,

avjl + /UV$0 + (// + b) D^V = 0

(/I + b) V Jj + /<V»V + aD^V = 0

On eliminating first V and then U between (5.1j), we reach

(5.1i)

JTLU = o, nv = o, (5.5)

U

15

the operator -fL being given by

ILm vh + aa-s^b-b2 VV + | A (S.6)

Consistent with the notation introduced in (2.2), we now define the

o operator V*" through

Vi = V, • <££ (i = l» 2) (5.7)

and seek to determine the coefficients cr, ct in such a way that

jfL=V2vf. (5.8)

To this end, we expand the right-hand member of (5.8) as a polynomial in

V?> « and equate its coefficients to the corresponding coefficients

appearing in (5*6). This leads to

.2.J. aa - 2b/U. - b2 2 2 _ a r- _»

^ 2 a^ ' 2 ' 2 2

so that c, , ct are the roots of the quadratic,

Q(£) = f+}£±m-=-&£ +-|-=o- <*-io)

The inequalities (3»lh) imply that Q( £) cannot have a negative or

vanishing root. Conditions (3«lU), however, do not preclude the exist-

2 2 ence of complex roots. Indeed, cT and ct are real and distinct, co-

alescent, or conjugate complex, according as

aa - (2/^ +b)2 JO. (5.11)

In the special instance of isotropy we conclude from (3.16), (5.10) that

o| * e| • 1, whence v\ « V2 (i • 1, 2) and !}_= V1*.

i

IXC

In view of ($.$), (5.8), and by virtue of the theorem proved in

Section 2, U admits the representation

(1) k (2) U = B + z B

9 (1) 0 (2) V£ B - 0, V| B =0

(5.12)

where

k =

0 if 4* 4 1 if 4 = 4-

> new functions A

> (2)

B (2) ,zz

V2 V2

(2) A : 0.

(5.13)

(l) (2) It is expedient to define two new functions Au>,z), A(p,z) by means of

(1) (1) (2) B = A + kA ,z ,z

2 (1)

v; A =o,

From (5.1M, (5.12), (5.13), (5.7) follows

r (1) k (2)-, U = De [A f z

k A)Z j

.2.(1). _2_k.(2) .,.2 (2)

-(5.1U)7

V^ = -^[^A • c- z~ LfZ - 2kcJ A ]

(5.15)

(5.16)

Substituting (5.15), (5.16) in the first of (5.U)» and performing two

successive z-integrations, we obtain

(1) , „ (2) (ac£ -/^)A + (ac^-/a)z

KA<

„ (2)

z

- 2aCgkA • 8 f, • » f, , (5.17)

where f. and f_ are functions of p alone. We first treat the general

case in which the degeneracy U. - - b is ruled out. - — _- _

'The existence of V ' and Av ' is readily verified with the aid of the last two of (5.12).

17

Case 1: n. • b ^ 0

In order to determine the nature of the functions f., f_, we substi-

tute (5.16), (5.17) into the second of C5-U)- This lengthy computation,

with the aid of (5.10), (5.13), (5.1U), leads to

V| fx = 0, vl*2 = o (5.18)

Now define 4>(p,z) by

,« T A3 i r (1) k (2>

a7;f3 = -2f2.

<p>], (5.19)

From (5.19), (5.15), (5.1.), (5.1, follows

u = - UJL • b) <b

uz = aV2^ + (/JL - a) 4?

(5.20)

and (5.13), (5.H), (5.18), (5.19) imply

= V72 „2 rt0 =vj^|4) =o. (5.21)

We have shown that, barring the exceptional case LL + b = 0, every

solution of the displacement equations of equilibrium (Ii.6) admits the

representation (5.20) in terms of the stress function <|) which is sub-

ject to (5.21). Moreover, a direct substitution confirms that every dis-

placement field of the form (5.20), with (J> a particular solution of

(5.21), satisfies (U.6) even if ^U + b = 0. Equations (5.20), (5.21),

except for an uressential constant multiplier of <p, are identical with

the stress-function approach given by Lekhnitsky [3]t and independently

arrived at by Elliot £7]. Applying the theorem of Section 2 to (5.21),

18

and recalling (5.7)» we note that the general solution of (5.21) may be

written as

$(p,z) = ({Xj-fy), z/c^ • zk4>2(/),z/c2), (5.22)

where (Jl(p,z), <P_(O,Z) are arbitrary axisymmetric harmonic functions

and k is given by (5.13)• The cylindrical components of stress belonging

to the displacements (5.20) are obtained with the aid of (li.3)> (h.U) and

may be omitted here.

In particular, if the medium is isotropic, let

x = 2/1 4>. 1-2^

In view of (3.16), Equations (5.20), (5.21) here reduce to

,2

(5.23)

2/*u = - X , 2/f^- 2<! " »V X- \ f zzJ

vhX. = 0, (5.2U)

8 which is Love's stress-function approach to isotropic problems with

torsionless axisymmetry.

Case 2; JUL + b = 0.

In this special instance (5.U) degenerate into

^^',B- 0, r^2v a 7

5.10),

A=%> c2-i- 2~yU'

= 0, (5.25)

(5.26)

See [1], p. 276.

19

Consequently, according to (5.1), (5.7)> the general solution of (li.6) now

appears as

u = U , u = V

V^U = 0, V^V = 0

and is thus again expressible in terms of two arbitrary harmonic functions.

•

Bibliography

Ql] A. E. H. Love, nA treatise on the mathematical theory of elasticity*, fourth edition, Dover Publications, New York, 19hh»

£z"2 J» H. Michell, "The stress in an aeolotropic elastic solid with an infinite plane boundary", Proc. London Math. Soc, vol. 32, 1900, p. 2U7.

£32 S. G. Lekhnitsky, "Symmetric deformation and torsion of bodies of revolution with aniaotropy of a special kind" (In Russian),, Journal of Applied Math, and Mech., USSR Academy of Science, vol. U, no. 3» 19U0, p. U3«

£1T] S. G. Lekhnitsky, "Theory of elasticity of an an1 so tropic body" (In Russian), Gosudarstv. Izdad. Tehn. Teor. Lit., Ifoscow- Leningrad, 1950,

£5] R. D. Mindlin, "Force at a point in the interior of a semi-infinite solid", Physics, vol. 7, 1936, p. 195.

£6T] B. Galerkin, "Contribution k la solution generale du probleme de la thebrie de l'elasticite dan's le cas de trois dimensions", Compt. rend. Acad. Sci. Paris, vol. 190, 1930, p. 101*7.

Q7] H. A. Elliot, "Three-dimensional stress distributions in hexagonal aeolotropic crystals", Proc. Camb. Phil. Soc, vol. UU, 19U8, p. 522.

QQ H. A. Elliot, "Axial symmetric stress distributions in aeolotropic hexagonal crystals. The problem of the plane and related problems", Proc. Camb. Phil. Soc, vol. U5, 19U9, p. 621«

£9|] R> T. Shield, "Notes on problems in hexagonal aeolotropic materials'*, Proc. Camb. Phil. Soc, vol. U7, 1951, p. U01.

£103 Ana Moisil, "Asupra unui vector analog vectorolui lui Galerkin pentru echilibrul corporilar elastice cu isotropie transversa'", Bui. Stiint., Ser. Mat. Pis., Acad. Repub. Pop. RomSne, vol. 2, 1950, p. 267.

£ll]] Gr. C. Moisil, "Asupra sistemelor de ecua^ii cu derivate par^iale lineare si cu coeficienji constan^i", Bui. Stiin^., Ser* Mat. Fiz., Acad. Repub. Pop. Rcri&ne, vol. 1, 19hy, p. 3bl»

£l2^ Emilio Almansi, "Sull' integrazione dell' equazione differenziale

^ = 0", Ann. di Mat., Ser. 3, vol. 2, 1899, p. 1.

20